6. Show that the followings define metrics on R2: For r = (11, 12), y = (y1, y2) ER, the company = 139-un +100 - 247 91.42.), y =(1,9) ER di(x,y) = |21 - y1| + |22 - y2), doo (I, y) = max{\21 – yı], \12 - y2|}.
(a) Show that the points (x1, yı), (X2, y2), ..., (xn, yn) are collinear in R2 if and only if 1 X1 yi X2 Y3 rank < 2 1 Xn yn ] (b) What is the generalization of part (a) to points (x1, Yı, zı), (x2, y2, 22), ...,(Xn, Yn, zn) in R'. Explain.
Rewrite each equation in polar coordinates. 19. r? + y2 = 25 20. x + y2 = 81 21. x = 12 Rewrite each equation in rectangular coordinates. 31. r= 5 cosa 32. r= 8 sino 33. r=7 Sketch a graph of the polar equation. -TT 47. r=4 50. 0= 3 51. r= 6cose
Let MM be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x2+y2=81, 0≤z≤1x2+y2=81, 0≤z≤1, and a hemispherical cap defined by x2+y2+(z−1)2=81, z≥1x2+y2+(z−1)2=81, z≥1. For the vector field F=(zx+z2y+4y, z3yx+4x, z4x2)F=(zx+z2y+4y, z3yx+4x, z4x2), compute ∬M(∇×F)⋅dS∬M(∇×F)⋅dS in any way you like (1 point) Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by X2 + y2-81, 0 < ž < 1, and a hemispherical cap defined by...
3) Let (x, y), (X2, y2), and (X3. Y3) be three points in R2 with X1 < x2 < X3. Suppose that y = ax + by + c is a parabola passing through the three points (x1, yı), (x2, y), and (x3, Y3). We have that a, b, and c must satisfy i = ax + bx + C V2 = ax + bx2 + c y3 = ax} + bx3 + c Let D = x X2 1....
3. Show tha the system of equations where r r2 + y2 has a limit cycle at r - To for each ro such that F(ro) = 0 and F,(m)メ0. The orbit is called asymptotically orbitally stable if F(ro) >0 and unstable if F(ro)0. For the case when F(r)r2)(2- 4r 3) find all limit cycles, determine the orbital stability, and sketch the orbits in the phase plane. 3. Show tha the system of equations where r r2 + y2 has...
please answer the question below Show that the set R2, equipped with operations (x1, y1)F(x2, y2) = (x1 + x2 + 1, y1 + y2 – 1) A: (2, 3) = (Ag+1 – 1, 2g - A+1) defines a vector space over R. Show that the vector space V defined in question 1 is isomorphic to R² equipped with its usual vector space operations. This means you need to define an invertible linear map T:V R2.
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that: 1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
2. Įpp. 492, Marsden & Hoffman Let y : [a,b] → R and ψ : R → R be continuous. Show that A = {(x,o(x)) : x [a,아 C R2 has volume zero in R2 and the set B-{(x, ψ (x)) : x E R} C R2 has measure zero in IK. 2. Įpp. 492, Marsden & Hoffman Let y : [a,b] → R and ψ : R → R be continuous. Show that A = {(x,o(x)) : x [a,아...
for b partfor b part Problem 2. The following non-linear system is given: y} = 11y2 + x3 - 1 yż = x2y1 +z1-1 (a) Determine if this system is solvable with respect to y = (y1, y2) locally around the point Z1 = (1,1,1,1) So, you have to determine if there are two functions yi = y111, 12) and y2 = y1 (11, 12) defined locally around x1 = (1, 1) and satisfying the given equations. If yes, determine...